For myself at least, it's not just having a succession of poor rolls that I don't like, it's having my actions be subject to the whims of the dice. It is, however, specific to my actions: I rather like randomisation of the playing field--both physical and metaphorical. For example, I don't like my hits and misses being determined entirely by die-rolls; I want my hits and misses to stem from my skill. (I don't mind a small element of random number generation; for example, I'd be happy enough if spells occasionally produced a more- or less- powerful effect, or a slightly different one, or if sword-hits were occasionally "critical" hits, etc.) On the other hand, I do like having randomly-generated levels, and randomly-generated loot; I like the idea, for example, of getting a sword on one run and a wand of fireball on the other. This sort of randomisation changes which options I have available, rather than changing the outcome of my selection from those options.

I like what Michael Brough did with ZAGA-33 and other games, in which the principle is that randomization of the world is enough, and otherwise you don't throw dice. (In practice, some enemy movement is random as well, but in principle you could do away with that as well if you have deterministic enemies that use the environment to decide where to go.)

My own thought is that for each source of outcome randomness (like throwing dice to determine combat resolution), the player should have an associated choice about how to mitigate the randomness. Equipment that avoids the negative effects of certain die rolls, but has other drawbacks, "bunting" stances that don't miss but don't critical hit, moves that leave you entirely open to attack but guarantee a hit in the next turn, anything that allows the player some say in how rolls are interpreted.

Speaking of card games, one thing I like about many card games is that no individual card is universally bad, but each card might be good or bad according to the choices I make. When the luck is against me, I can't entirely blame the cards, because had I chosen a different strategy that card may have been exactly what I needed. Likewise, if I had choices w.r.t. randomness mitigation, then when the bad "roll" happens I can't just say "this sucks, I'm so unlucky", I say "oh, that's partially my fault, I could have taken the other choice".